Parallelotope tilings and $q$-decomposition numbers

We provide closed formulas for a large subset of the canonical basis vectors of the Fock space representation of $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_e)$. These formulas arise from parallelotopes which assemble to form polytopal complexes. The subgraphs of the ${\mathrm{E}\mathrm{x}\mathrm{t}}^1$-quivers of $v$-Schur algebras at complex $e$-th roots of unity generated by simple modules corresponding to these canonical basis vectors are given by the $1$-skeletons of the polytopal complexes.